[Music] so [Music] oh me [Music] do it [Music] [Applause] [Music] [Music] so [Music] [Music] [Music] [Applause] [Music] [Music] so [Music] [Music] you [Music] [Applause] do [Music] [Music] [Applause] [Music] so [Music] [Music] uh [Music] to the interaction between the micro microscopy constituent and and since uh quantum theory is is is what we use to describe uh [Music] and another approach the external local magnetic field and the g as the interaction between the spins this model is exactly solvable in one dimension while it is the case in two dimensions uh only for zero magnetic field [Applause] does not have a finite uh phase transition in one dimension we have a ferromagnetic two-part little phase transition uh which can be described uh by the magnetization phase transitions are not exclusive to finite temperature settings as we can find a whole range of critical phenomena at zero temperature which we call quantum phase transitions which are driven by non-terminal parameters such as the magnetic field similarly with the classical counterparts quantum phase transition can be either first or second order phase transitions and a first order quantum phase transition emerges as a concept as a consequence of energy level crossings at a critical point gc why in second order quantum phase transition [Music] occur due to avoided energy level crossings at gc quantum phase transitions have been verified with various architectures such as charcoal atoms and quantum dots and in order to describe the critical properties of quantum systems we have a simple model which is the quantum analog of the easy model where now the quantum degrees of freedom are given by all the polyspin matrices and external magnetic field now quantum many body systems are special since at zero temperature we can describe them with a complex ground statewide function which contain all the relevant information about the correlations which are responsible for the emergence of various phases of matter such as superconductivity and magnetism and these correlations we can capture them using quantum information theoretic measures from quantum information theory therefore are motivated to study quantum critical phenomena using the quantum information approach and we will do this by representing quantum states you are using density matrix in order to define quantum correlations and we will also use another equivalent representation of quantum states through the invignal function in phase space the most famous quantum correlation is the impediment which go way back to 1935 where einstein podeski and rosen introduced the notion of intended state and so if we have two particles and they are intended we cannot describe the properties of the left particle independently of the right particle and this non local property of the entire element was used in various applications such as quantum communication quantum cryptography and teleportation which were verified experimentally in order to measure and quantify entanglement there are various quantifiers and in our case we are interested in qubits and waters evaluated uh the entanglement between two qubits within the conference where the lambda eyes represent the eigenvalues of the phase flipped version of our our density matrix the problem with the entitlement is that it quantifies the correlations only from the perspective of separable and non-separable states and we can find a whole range of other general quantum correlations and to capture them the muscle of the quantum discord was introduced by vedro and zurek which quantifies quantum correlations by the difference between total correlations measured by the immediate information by iad and classical correlations with this quantity j a b which quantifies classical correlations by the difference between the entropy between the subsystem a and the entropy the conditional entropy of knowing system a while having some information about system b the origin of quantum correlations seems from the notion of coherence which is a central concept in physics and to measure coherence there are various measures either entropic or distance measures and in this tool i will be we will be interested in a special measure of coherence which combines the metric and entropic property which is the quantum version of the jensen divergence where ro is our system of interest and sigma is the closest incoherent state to which in this case we take it as the diagonal version of our density matrix problem so now that we've got all this background we are ready to apply it to classical quantum systems and we consider the xy model which describes a lattice of spin halves where g is the interaction between the spins gamma is the anisotropic parameter and h is the external magnetic field and without those of generality we consider the x x limit which is taken by considering gamma equals zero and this system has a second order function between hc equals one in order to evaluate the quantum correlations we consider an infinite chain and using the property of the trace operation and body basis expansion we can find the reduced density matrix of only two qubits as given by equation four and we are interested in this with situation of long-range quantum correlations such as correlation between the first site and its second third and fourth neighbors so we start with reporting the results of the entanglement and blocks in the temperature magnetic field uh plane the entanglement uh from from the left to the right of the second third and fourth neighbor and we see that regions of entanglement are small and get smaller as the distance between the spins gets bigger and we see that the entanglement is maximum exactly at the at the transition point between the two phases and we report uh the weakness of the entanglement uh controller fluctuations in contrast uh quantum discord is defined in the whole in the whole plane which is which is expected since only the coherence kills quantum discord and similarly we find that quantum discord is maximum at the transition point by the entanglement but here we report the robustness of quantum discord to terminal fluctuations the behavior of control coherence is similar to the one reported for quantum discord which is extracted since quantum discord originates from the coherence between the subsystems but here we report a more resilience through thermal fluctuations than quantum display for the entity so in order to show that quantum information theoretical measures can be used uh to deduct the critical properties of quantum systems uh to consider the derivative of the entanglement uh quantum discord and the quantum coordinates where we see that they diverge for all the cases which means that they successfully detect the second order complicated in this in this moment so now that we have explored the quantum correlation path we are really uh to study the face place approach so what's the history with with bass space uh in face space the player is no longer the winner function but uh he does the matrix but instead it is a positive probability distribution called the beginner function and equation five represents the beginner function for inferior state which can be written in in general in general for for any system as the the average of the density matrix go in a kernel delta where data can be written as in terms of the displacement operator the d and the parity operator the wigner function is considered a positive probability because it can be negative numbers and this can be seen by the following two examples where we show on the left the regular function for iphone state and on the right we show the beginner function for a car state and we see that they take negative numbers and the negativity is often the negativity of the lunar function is often associated with the known classicality of our quantum state and it was shown that the female function can detect the entanglement in quantum systems and we have just seen that the entanglement can be used to uh to detect controversies transitions in critical quantum systems so the natural question that we ask here is whether it is a religious relationship between the indignant function and from this transition and in particular whether the female function itself can detect critical properties of quantum systems so here we introduce our new approach of studying common phase transitions in the phase space and to achieve our goal we need a bigger function for quantum systems and in particular we need a suitable parallel data in order to describe discrete quantum systems and in the literature there are many attempts in in generalizing the vignette function for quantum systems but here we are interested in a particular construction which was defined by by waters back in 1997 and you have chosen this particular construction because it works well with speed systems but it only works for uh for systems with the with a prime prime the dimension and in this construction the phase space for one qubit consists of four points and the first phase is labeled by x and p uh where they take zero or one and for two qubits uh the phase space consists of 16 points and each point is described by a phase operator a as given in equation seven in terms of polystyrene matrices so that the root as uh beginner function can be written as the average of the density matrix law in the in the independent a so now we are ready uh to apply this this formalism to a single system in order to see if it works and we call again for our x x y model but this time uh we will uh we will uh we will consider non non zero uh isotropic parameter in this case we have a second order pointing this transition at the critical field lambda c equals one and we have a second critical phenomena which is the factorization of the ground state at the critical field lambda f equals one over the square root of one minus gum square and this factorization of the ground state can be seen by uh concurrence in the two cubic of the xy model which we plot in this in this figure on the right in the gamma lambda plane where the white line is the is the factorization line in the xy model in the following we consider the damage to 0.5 which means that the factorization of the ground states of the ground state occur exactly at lambda equals 1.50 and we will apply this to our reduced density matrix for uh to qubit so that the wigner function can be written in terms of the magnetization and the space spring correlation functions in the x y and the z direction the phase space consists only of six behaviors which get repeated over the base space and we calculate directly the derivative of all these six behaviors and we plot we plot each behavior with its corresponding symbol in phase space and we see that the wigner function can detect the second order of the base transition by showing a divergence at lambda equals one but we still missing the factorization phenomena it was one point fifteen in order to detect the factorization of the ground state we consider the derivative of the regular function that is based on the square root of the density matrix and when we do this we see that the variable function for all the sex behaviors takes a jump exactly at lambda equals 1.15 next we consider the long-range behavior that is the beginner function for the first site between the first site and its 20 neighbors and we report similar behavior to the nearest case um exactly when we take the density matrix or is its square root so now that we know our our method works for simple systems we are ready to apply it to a more complex system that is the xxz movie and this uh model have two phase transitions uh one at delta equals minus one where a first order phase transition occur and at delta equals plus 1 there is a topological quantum phase transition the phase space in this case consists of three behaviors which again gets repeated over the whole face place and we plot each behavior with its corresponding symbol in phase space and in the inset we show the first and second derivative of the linear function and in this first behavior we see a clear discontinuity of the wooden function after that equals one which means uh it detects easily the first order point of this transition and the regular function is maximum in the negativity as delta equals one and the derivative diverges in in the first order continuous transition at the level of the second behavior we report a similar behavior of the first behavior but here we see a small bump around around delta equals one at the level of uh at the level of the topological phase transition only in the third and final behavior where we see some indication of the topological control phase transition around delta equals one so so far we have been able to detect uh only the first order point of this transition and we are still missing the other topological quantum based transition and it was shown that only quantum discord was able to detect efficiency efficiently this topological content-based foundation and since quantum discord involved the externalization operation we were motivated to propose the maximization and the minimization of the ignorant function and we do this by maximizing and minimizing over the three prominent behaviors in phase space and when we do this at the level of the mat which is the plot of the plot above we see a clear kink exactly at uh the topological content phase transition and at the level of the second directive which we which we plot in the inset uh we see a clear divergence exactly at delta equals one which means that the video function can successfully and easily detect the topological uh quantum phase function at the level of the minimum of uh divergent function which is something interesting that is we see that the concurrence is proportional to the absolute value of the minimum of the vignette function so with this this approach we showed that how the bigner function can easily detect first second and extract the information about the entanglement and the xxd model and these results takes a first first step in order to propose a universal quantifier for all uh phenomena so now we have been only focusing on equilibrium properties of critical quantum systems and as christopher said if we shift our focus away from equilibrium states we find a rich universe of non-equilibrium behavior and here we are motivated to study non-equilibrium quantum based solutions using the thermodynamics approach because of the tight connection between information and thermodynamics information can be measured through the entropy which is a central concept in statistical physics which is the backbone of the theory of thermodynamics from the other hand there is a minimum thermodynamical cost for processing information through the landlord limit which means that information is a physical quantity so in this part we are motivated to study how work entropy and information behave upon crossing a non-equilibrium only case transition and we will not rely on traditional thermodynamics is it is an equilibrium theory but we move to stochastic thermodynamics which treats new equilibrium settings where work now follow a probability distribution as given by equation 12 and stochastic due to through stochastic thermodynamics many fluctuations fluctuation theorems were invented such as the kazakh books theorem and the azimski quality so next next we will apply our approach to the lmg model which is a collective spin model and can be considered as the infinite connecting limit of the easy mode where the sis are the collective spin operators h is the magnetic field and l is the number of spins this model have a second order continuous transition between a ferromagnetic phase where the energy levels are are degenerate and a paramagnetic phase where the energy levels are equidistant and the transition occur at the vertical field of h equals one in the ferromagnetic phase we have an interesting phenomena at the excited state which which is called excited state quantum phase transition which is due to the lifting of the the degeneracy and the energy levels and this can be seen by looking at the energy levels of uh of the amg model where we've got here a part of the energy levels of the a hundred sites in the energy model where we see that at uh at the critical line of e equals zero the energies transits from being degenerate to non-degenerate and this crossing of of this line appears a as a non-anonymous density of states that is due to the concentration of all the energy levels around e equals zero so in this part we are interested in studying how crossing this excited state of this transition affects the work statistics of the lmg one and the protocol we consider is we will initialize our system in the ground state psi zero at a given value of the magnetic field which is 0.5 and we will study three scenarios first we will quench uh below then at and above the excited statement phase transition where the critical field of the excited state contin 0.75 and we are interested in three figures of merit which the first is the rosh medical or the survival probability which measures how our initial state psi zero resembles the final state of psyche after the question and you are interested also in the work probability distribution and it's shannon entropy so first we start with initially initializing our system in the symmetric ground state and we plot for a system size of two thousand spins the loss net effort with a structured line and we see when we quench above and below the physical point you see these revivals which are distant by periods of orthogonality and when we quench exactly to the excited statement of this transition the system remain in the orthogonal state this behavior uh is reflected in the uh what probability distribution where uh when it is um where it has this uh reduction for above and below the excited state phase transition and when we quench to the critical point we have this double peak in the probability distribution at the level of the channel entropy we plotted for several system sizes versus the final magnetic field where the channel entropy shows a heat exactly at the critical field of the excited state quantum phase transition and again we show the maximum of the channel entropy with respect to the logarithm of of the system sizes and where we see this linear dependence so next we initialize our system from a symmetry broken ground stage and we do this by adding a small term in the direction x in the in our hamiltonian and we plot again uh the loss multiple versus time for a system size of 2000 spin and the difference we see with the symmetric case is that above quenching above the excited signal of this transition we now have fewer revivals at the level of the work probability distribution uh we see again that the work distribution is gaussian above and below the vertical but when we quench to the excited state phase transition we see that the symmetry of the wp now breaks and at the level of the channel entropy we report a similar behavior to the symmetry case but now we have higher values in the initial entropy and now we will initialize our system in a superposition between a symmetric and fully symmetry broken ground state and when we do this uh the superposition can case or lie exactly between the symmetric and fully symmetry breaking case and a similar feature is reported also for uh and both plots are for a system size of 2000 spaces finally we quench from an excited state and why the behavior is similar to the ground state case here we see that we have more frequencies and the losses echo and at the level of the probability distribution will be quenched exactly to the excited state based transition we see this being modality in the world distribution so through this presentation we have uh we have discussed various concepts from correlations beginner function and work statistics in order to show how all these approaches can be used to detect critical features of quantum systems and so these results are shown in this presentation were published in three papers and in three papers in international journals and were presented in various conferences in morocco italy austria portugal and japan for the future the results we found for the interplay between the excited state conference transition and work statistics motivates me to study how we can use critical phenomena in order to enhance efficiency of quantum terminal machines in particular how we can how we can model a quantum thermal machine to a quantum system which is driven across its excited state from base transition and how this driving will help with the efficiency of quantum terminal machines for the far future i'm interested in studying energetic advantage of quantum computation uh especially quantum animals and also i'm interested in studying tensor networks was called simulative manipulating physics and this second project is motivated by the recent push for a quantum energy incident and thank you for your thank attention very much [Music] to start with the person who are far from our university but supervisors will be given to discuss so let us name correctly from the institute of theoretical and applied informatics so yes please professor graham uh yes around it's a very constant process in the small world so do you have any practical reasons for example practical realization of the easy no i'm afraid not so [Music] the information is so let's go to professor mohammed from university thank you very much thank you thank you during my vacation reading it and i was very happy to write a report on it so can you please go to uh to slide 11. okay so here you talk in general using these techniques for for example detecting quantum criticality for example systems so you have the xx model for example and you can see there are the case gamma equal to zero so can you can you say something about another parameter because when we have a classical phase transition but here can you can you talk about the equivalent of the monetizations in this case [Music] is [Music] is [Music] [Music] here is brought by the xy components of the spin right yes by the quantum way with which we represent spins in here it's the volume matrices but the classical version is constant and it's just the magnetization say for example if you consider suppose you consider the eisenberg the full one form one yes xyz yeah isotropic okay yeah and then you apply magnetic field in one geek we know that there is magnetization saturates at a magnetic field which is equal to two times [Music] [Music] easy so could this technique say something about yes yes you can apply it yeah yeah and just take this density matrix and the averages if you know how to calculate the spin speed correlation functions you can calculate the concurrence and the other uh continuous not what's also interesting about your thesis is [Applause] physics can be [Music] knew how to represent things from from face space to this e matrices but the inverse uh we didn't know the inverse of operation so we can go from face space to row to divide transformation but we don't know how to go from from from role to to face space and vignette invented the inverse of of the transformation so this is the story about it he invented the function but it gained interest only in the 50s in quantum optics and that stuff [Applause] [Music] [Music] [Music] to them that's why it's hard to to detect them yes in this case uh we showed that using the figure function we can spot it easily so this is what we did in this in this world but there was point in this quote it's it's uh it's a hard to evaluate express because because you need to take the the minimum operation but here with the wigner function taking the minimum is an easy uh the maximum is an easy task this is what we showed in in our paper so that's a delta one yes yes and these techniques may help yes with this kind of transitions can they happen in other systems systems for example i think they occur in the uh model but i don't know i don't know other homes or cube rates high tcc superconductors there is the idea that the hidden phase transition may exist these materials so put these techniques out you can if you can define a density matrix then you can apply uh apply the bigger function in this you can apply apply the function for them and follow the cube rays for example they are they they have both degrees of freedom spins and charges so you need to take them into account it's not like the spin mode we can use a general construction for the wigner function which i show in the manuscript and indicator but i didn't present it here they started with each function it applies to every quantum system once you have your density matrix row you plug it in the webinar function and you start doing calculations so whether whether the both degrees of freedom are present yes it doesn't need a density matrix once you define role that's it oh thank you very much professor of the professor for your interesting discussion uh just before i would like just to ask professor scandal and to follow the entire discussion is it okay now it's better is it okay okay yeah this material science and engineering so i would appreciate if you uh if you can give us some some examples from some materials and you say that those are transitions this is how we reach temperature for example we are talking about [Music] [Music] you say that this applies for this kind of material i don't [Music] spin moments another aspect was about first slide how the temperature of filtration is impacting for example uh first and then it easily destroys the [Music] [Music] [Music] and what this means is that you have a product statement with with this chord the situation is different so even if you have a product state it still have quantum correlations it's not the case once once you once you separate these two it's [Music] [Music] [Music] [Music] when you consider the contact between the system [Music] [Applause] um you said that i show another another method which is more general it can apply for any quantum system so once you take your computer system you describe it with the density matrix rule and you plug it in the signal function the difference between this method and the water's case is in the kernel i don't know i can't you can say so this is this is what's uh what's about with the function you need it only works for uh for specific systems so you need to apply it first then you will know okay so from your perspective did you look into this for [Music] so what i'm interested in is to apply what we did with the energy model in the case of of the easy spring glass which we use in order to encode the solution of a problem this is this is how quantum animal works so thank you very much professor mr from international university do you want to add something to comments again we thank you very much president i would like to dust with the foreign [Music] is um [Music] the is foreign [Music] foreign but uh there is a way to uh to extract that the exponents from from the vegan function and i'll show you why so here from this equation uh if you take specific values of x and b you can end up with a situation where the wigner function is proportional only to the magnetization and we know how to start and we know how the magnetization diverges at the critical point and from that we can we can extract the exponent beta since dividing that function is just proportional to sigma z in some cases foreign construction you can do it with the other construction that i record in the manuscript and also in the paper it's called the saturn which treatment function it works for all quantum systems once you define the density matrix for your system and you can use the vignette functions foreign [Music] changing the type of interaction doesn't change the properties of your model what you see for for the ferromagnetic case can be expanded to the only terminal case so they are similar um [Music] perspective foreign [Music] situations i think you need to change the kernel uh which you you generate the volume function and again your your density matrix also changes uh foreign [Music] you can define a density matrix for that particular system and you plug it in the individual function it works for every every situation but not waters but the certain of which yeah it is it could not count directly now you can add it also because rock is just exponential [Music] m foreign so first of all presentation so in your presentation i have two nice concepts the first one is similar to the second one so my question is the configuration of the sections in your faces are not simple because the first thought that contained two sections the second subject contains six sections i i would like to see simply in chapters so this is the very important answer you should i'm [Applause] of scene do you have okay so it's a continuous series of this excellent because the the kind of thing that is very important okay so the lmg model has the spin flip symmetry all right is very complicated and business because we should talk about topographic environments and so where this topology comes from you know or all we can find topology when you're in your physics you know because you know the physical topology is very very complicated it's not like a genetic one which is for students well we only studied uh topology in the xxc model in terms of whether we can detect it or not we didn't go into [Music] in category because that's that's itself from that so so wish you a good uh good luck in here [Music] from university college of dublin supervisor of the thesis i hope you are you hear me yes please professor stevens um me and we established uh uh our incredible lineup of work and introduced me to this this space-based approach to the bigger discrete systems that i had never ever considered um i think what impressed me was they came with a very very clear ideas and then um i think you know we had some some grandparents about interrupting um [Music] [Applause] highlighting as well is that one of the things that really emerged amongst the collaboration winston and the work to establish much more factual connections i think between dublin and uh between ireland and morocco so obviously at this point i don't think there have been a huge amount of collaboration or a huge amount of overlapping uh [Applause] throughout the uh and i'm hoping that this is just the start of a productive career i think with that i i just congratulate you again so uh thank you very much professor for your all your comments and also for your convenience so now uh i would like to go to professor mark the best room and i i thank strongly for his choice to this beautiful jewelry that makes up the top the evolution of these of these days so first i would like to thank all the members of the jury for accepting to be among us today so it is pleasure and an honor to have you all or with us especially to start with the reporters because we had to work under pressure i mean we wouldn't be able to uh to make it i mean with these big lives and all this without without the reporters giving back the report same time and i'm sorry for explaining mr azuz he had to work during the vacation to university i enjoyed everything thank you very much um i would like to thank professor gaddis for accepting the to be with us here also we would like to have them both steve and uh and batek with us with us here but unfortunately uh with all the circumstances it is not possible but we enjoy nonetheless that we are here online so thank you all for being uh for being here today um i mean it was it was pleasure from from the beginning working with with with zakaria it's today is the culmination of course of the entire period we you had to work hard for the entire period of your thesis we we were able to and i'm glad to say this that we were able to surpass our expectations the limits that we that we put at the beginning we were able to surpass them and that's a great part thanks to professor steve campbell who accepted to help you and to supervise the thesis it was it was pleasure and all this is of course thanks to to [Music] it is thanks to you that i have to meet steve and i have to meet bartek so i would have to thank you also for the for all this we mean we still have a lot of work uh to do under the progress but uh with all this uh uh with all this as i said we were already able to surpass our the points that of the ideas that we wanted to discuss that we at the beginning now um here i i was just not in some of the uh uh the points that were raised during the the discussion and i i thank the all the members of the jury for this beautiful discussion i really enjoyed being part of this this jury and having you with us for this uh for this discussion most of the uh comments uh were about um uh journalizing the technique uh of phase space to to other systems we we have some work and the progress with with the career and other other students to see how this how this works the idea that was just to test it in simple simple models and then to see [Music] what it can do in more complicated more complicated models so far to since promising when we are when we are going to rise the dimension of the systems that's one of the difficulties using working with the usual techniques the use of these phase based techniques seems since practicing seems promising perhaps not the waters way of doing things but the marginalized one since seems easier so i mean despite zakaria's commitment to finish his thesis in this period he is still working and he's also helping the other students in their in their works so that also shows the level of commitments that zakaria gives to to his work now i think there was a remark about symmetry for uh adid that's a problem with the with zakaria he loves symmetry breaking so it's not you cannot expect to have symmetric things some of the things we one of the remarks that we use that we had with them is just for the diagrams they have never symmetric so that's it again i would like to thank all the members of the jury thank zakaria for the wonderful the period of this of this series so thank you mr president thank you very much it's not sure that you get what is this [Applause] uh thank you all for coming here and i'd like to thank you for their time and for their remarks and also i would like to thank my supervisor for that time and for uh for supervising me and thank you so awesome is [Applause] oh [Music] [Music] hello [Music] so [Music] [Music] you